Asymptotic mean value formulas for parabolic nonlinear equations

TL;DR

This paper characterizes viscosity solutions to nonlinear parabolic equations, including Monge-Ampère types, using asymptotic mean value formulas linked to probabilistic interpretations and game theory.

Contribution

It introduces a novel asymptotic mean value characterization for viscosity solutions of nonlinear parabolic PDEs, connecting PDE theory with probabilistic and game-theoretic approaches.

Findings

Viscosity solutions are characterized by asymptotic mean value formulas.

Formulas have a probabilistic interpretation via Dynamic Programming Principles.

Applicable to a range of nonlinear parabolic equations, including Monge-Ampère.

Abstract

In this paper we characterize viscosity solutions to nonlinear parabolic equations (including parabolic Monge-Amp\`ere equations) by asymptotic mean value formulas. Our asymptotic mean value formulas can be interpreted from a probabilistic point of view in terms of Dynamic Programming Principles for certain two-player, zero-sum games.